Trigonometry Examples

$x (2 x - 5) = 12$

Step 1

Simplify $x (2 x - 5)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x (2 x) + x \cdot - 5 = 12$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 5 = 12$

Step 1.1.2.2

Move $- 5$ to the left of $x$ .

$2 x \cdot x - 5 \cdot x = 12$

Step 1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1

Move $x$ .

$2 (x \cdot x) - 5 \cdot x = 12$

Step 1.2.2

Multiply $x$ by $x$ .

$2 x^{2} - 5 \cdot x = 12$

$2 x^{2} - 5 x = 12$

Step 2

Subtract $12$ from both sides of the equation.

$2 x^{2} - 5 x - 12 = 0$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = - 5$ .

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Step 3.1.1

Factor $- 5$ out of $- 5 x$ .

$2 x^{2} - 5 x - 12 = 0$

Step 3.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$2 x^{2} + (3 - 8) x - 12 = 0$

Step 3.1.3

Apply the distributive property.

$2 x^{2} + 3 x - 8 x - 12 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + 3 x) - 8 x - 12 = 0$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 3) - 4 (2 x + 3) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$(2 x + 3) (x - 4) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 3 = 0$

$x - 4 = 0$

Step 5

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 5.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 5.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 5.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7

The final solution is all the values that make $(2 x + 3) (x - 4) = 0$ true.

$x = - \frac{3}{2}, 4$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}, 4$

Decimal Form:

$x = - 1.5, 4$

Mixed Number Form:

$x = - 1 \frac{1}{2}, 4$